We examine the stellar velocity dispersions σ of a sample of 48 galaxies, 35 of which arespirals, from the Palomar nearby galaxy survey. It is known that for ultra-luminous infraredgalaxies ULIRGs and merger remnants, the σ derived from the near-infrared CO bandheads is smaller than that measured from optical lines, while no discrepancy between thesemeasurements is found for early-type galaxies. No such studies are available for spiral galaxies– the subject of this paper. We used cross-dispersed spectroscopic data obtained with theGemini Near-Infrared Spectrograph, with spectral coverage from 0.85 to 2.5 µm, to obtainσ measurements from the 2.29 µm CO band heads σ CO and the 0.85 µm calcium tripletσ CaT. For the spiral galaxies in the sample, we found that σ CO is smaller than σ CaT, with amean fractional difference of 14.3 per cent. The best fit to the data is given by σopt = 46.0 ±18.1 + 0.85 ± 0.12σ CO. This ‘σ-discrepancy’ may be related to the presence of warm dust,as suggested by a slight correlation between the discrepancy and the infrared luminosity. Thisis consistent with studies that have found no σ-discrepancy in dust-poor early-type galaxies,and a much larger discrepancy in dusty merger remnants and ULIRGs. That σ CO is lower thanσopt may also indicate the presence of a dynamically cold stellar population component. Thiswould agree with the spatial correspondence between low-σ CO and young-intermediate-agestellar populations that has been observed in spatially resolved spectroscopy of a handful ofgalaxies.Nota general